Edge Selection - Selection highlights vertices only?
Looking at your image, you are in vertex selection mode, that doesn't explain why the edge in between the two selected vertices isn't highlighted, so most probably you have doubles. Select all by pressing A once or twice, then  W  and choose remove doubles, watch the status bar at the header of Blender to see how many vertices are removed, this should work normally, you can set selection by edges instead of vertices by pressing Ctrl +  Tab  then choose Edge.